An Unconditional BGST$\to$R$_2$ Fourier Transfer via Poisson-Kernel Deconvolution

Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh [BGST23] proved the first unconditional asymptotic for Montgomery's pair-correlation function \(F(\alpha, T)\), with error \(O(1/\sqrt{\log T})\) uniformly for \(\alpha \in [0,1]\). We show that their result transfers unconditionally to the bandlimited pair-correlation sum \(R_2(T; w) = \sum_{\gamma\ne\gamma'} w((\gamma-\gamma')\frac{\log T}{2\pi})\) for every Schwartz test function \(w\) with \(\operatorname{supp}\widehat{w} \subset [-1,1]\), yielding \[R_2(T; w) \;=\; \frac{T\log T}{2\pi}\!\int_{\mathbb{R}}\!\Bigl(1 - \bigl(\tfrac{\sin\pi v}{\pi v}\bigr)^2\Bigr)\, w(v)\, dv \;+\; O\!\bigl(T\sqrt{\log T}\,\|\widehat{w}\|_{L^1}\bigr).\] This matches Montgomery's 1973 conditional formula with no RH assumption. The transfer mechanism is a mass-one Poisson-kernel deconvolution of the Montgomery weight, with relative error \(O(1/\sqrt{\log T})\) inherited losslessly from BGST. As applications we derive unconditional zero-gap statistics. Numerical verification against 100,000 Odlyzko zeros confirms the identity to six significant figures.

Keywords: Riemann zeta function, pair correlation, Montgomery conjecture, BGST theorem, Fourier transfer, Poisson-kernel deconvolution, zero gaps.

MSC 2020: 11M26 (Nonreal zeros of \(\zeta(s)\)), 11M06 (\(\zeta(s)\) and \(L(s,\chi)\)), 42A38 (Fourier and Fourier–Stieltjes transforms).